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Variant of the Maximum-Modulus Principle

  1. Nov 11, 2009 #1
    1. The problem statement, all variables and given/known data
    Show that the maximum principle holds with [tex]\phi :\mathbb{C}\rightarrow\mathbb{R}[/tex], 

    [tex]\phi(z) = (\textrm{Re}(z))^4 + (\textrm{Im}(z))^4[/tex],

    in place of the modulus:

    If [tex]U \subset \mathbb{C}[/tex] is open and connected, [tex]f : U \rightarrow \mathbb{C}[/tex] is holomorphic and [tex]p \in U[/tex] is such that
    [tex]\phi(f(p)) \geq \phi(f(z)) [/tex]

    then [tex]f[/tex] is constant.

    Can one also replace the modulus by the sine of the modulus, i.e. the function with [tex]\phi(z) = sin(|z|)[/tex]?

    2. Relevant equations
    If [tex]z=x+iy[/tex]


    3. The attempt at a solution
    I tried using the open mapping theorem and the fact that

    [tex]f(U)\subseteq\{z|\phi(z)\leq\phi(f(p))\}=\{z|x^4+y^4\leq M\}[/tex]

    where [tex]M=\phi(f(p))[/tex], but was not successful.
  2. jcsd
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